The observation that the human mind operates in two distinct thinking modes--intuitive and analytical- have occupied psychological and educational researchers for several decades now. Much of this research has focused on the explanatory power of intuitive thinking as source of errors and misconceptions, but in this article, in contrast, we view…

Motivated by the question, "What exactly about a mathematical concept should students discover, when they study it via discovery learning?", I present and demonstrate an interpretation of discovery pedagogy that attempts to address its criticism. My approach hinges on decoupling the solution process from its resultant product. Whereas theories of…

In this paper I argue my opposition to the consensus which has dominated the literature that young children view shapes as a whole and pay no attention to shape structure and that geometrical thinking can be described through a hierarchical model formed by levels. This consensus is linked to van Hiele's weok by van Hiele-based research. In the…

In this paper I take a positive view of ambiguity in the learning of mathematics. Following Grosholz (2007), I argue that it is not only the arts which exploit ambiguity for creative ends but science and mathematics too. By enabling the juxtaposition of multiple conflicting frames of reference, ambiguity allows novel connections to be made. I…

In this paper, we engage in a thought experiment about how students might notate their reasoning for composing fractions multiplicatively (taking a fraction of a fraction and determining its size in relation to the whole). In the thought experiment we differentiate between two levels of a fraction composition scheme, which have been identified in…

Current conceptualizations of knowing and learning tend to separate the knower from others, the world they know, and themselves. In this article, we offer "relationality" as an alternative to such conceptualizations of mathematical knowing. We begin with the perspective of Maturana and Varela to articulate some of its problems and our alternative.…

Discusses a class on subtraction or difference-finding, problems such as "There are eight white flowers and five red flowers, how many more white flowers are there than red flowers?" used in the teaching of Japanese first grade children. Describes three instances of introductory teaching of "difference-finding" problems in the first grade.…

Presents a study to describe and understand the thinking processes of students in a computer environment while undertaking mathematical tasks related to the differentiation of functions defined on real numbers. Describes two different approaches, the visual and the algebraic approach, in the thinking processes of calculus students. (Contains 19…

Presents a case study that aimed to obtain information on students' mathematical comprehension levels and on whether students may or may not make use of visualization processes in solving mathematical problems. Discovers students' beliefs about teaching and learning processes in general, and mathematics in particular. (Contains 25 references.)…

Interviews students majoring in mathematics who had passed a required introductory course on algebraic structures on students' difficulties with basic concepts in group theory as part of a research project. Reports data concerning cyclic groups. (ASK)

Explores the mathematical practices of three young mathematicians in an extended interview setting. Focuses on the interaction of discovery and verification, the role of conjecture in discovery, and the place of intuition and understanding in research. Indicates an interesting mismatch between how they valued their own guesses and how they reacted…

Principles on which the teaching of mathematics is based are discussed. Sections concern the skill principle, the curriculum principle, and learning strategy, with many classroom illustrations. (MNS)

The conceptual difference between understanding and algorithmic performance is examined first. Then some dilemmas that flow from these distinctions are discussed. (MNS)

How students are convinced that they have the correct solution to a problem, free of contradiction, is discussed. The role of counterexamples and the need for a situational analysis of problem-solving behaviors are each included. (MNS)

Some pedagogical problems in Chinese numeration are described. They involve the teaching and learning of how to speak numerals with fluency in Chinese, using Hindu-Arabic written numbers. An alternative approach which stresses regularity is proposed. (MNS)